Equilibrium Mean-Field-Like Statistical Models with KPZ Scaling

We have considered three different “one-body” statistical systems involving Brownian excursions, which possess for fluctuations Kardar–Parisi–Zhang scaling with the critical exponent $$\nu = \tfrac{1}{3}$$ . In all models imposed external constraints push underlying stochastic process to a large deviation regime. Specifically, we for: (i) excursions on non-uniform finite trees linearly growing branching originating from mean-field approximation of Dumitriu–Edelman representation matrix models, (ii) (1+1)D “magnetic” Dyck paths within strip width, (iii) inflated ideal polymer ring fixed gyration radius. latter problem cutting off long-ranged spatial and leaving only “typical” modes stretched paths, ensure KPZ-like bond fluctuations. To contrary, summing up normal modes, get Gaussian behavior. KPZ emerge in presence two complementary conditions: trajectories are pushed region phase space, leaning an impenetrable boundary.